Shapes of graph and eigen vectors

Introduction

The shapes should affect on the eigen vectors.

We expect that the symmetricity should strongly affect on the appearance of eigen vectors along with their corresponding eigen values.

Plans

• Make a simple graph and calculate their eigen vectors.
• Combine/Connect multiple simple graphs and calculate their eigen vectors.
• Attempt to identify relation between eigen vectors and the connection pattern.
• Initially we should start flat grphs, so that we can relatively easily identify the rules if any. 基本形ltticeを作成（座標とグラフオブジェクトとしての情報）

Strategies

• Generate latties
• Connect lattices
• Eigen-decompose their adj matrices
• Color the graph with eigen vector element values

Utility functions

my.col.st <- function(x,N=5){
  floor((x-min(x))/(max(x)-min(x)) * 0.99 * N)+1
}

my.color.point <- function(X,col,cex=0.5){
  n <- length(unique(col))
  ucol <- unique(col)
  for(i in 1:n){
    tmp <- which(col==ucol[i])
    points(X[tmp,],pch=20,col=ucol[i],cex=cex)
  }
}

Lattice in the shape of regtangle

my.rect.lattice <- function(xy){
  X <- 0:xy[1]
  Y <- 0:xy[2]
  XY <- as.matrix(expand.grid(X,Y))
  n <- length(XY[,1])
  d <- as.matrix(dist(XY,method="manhattan"))
  adj <- matrix(0,length(d[1,]),length(d[,1]))
  adj[which(d==1)] <- 1
  return(list(X=XY,n=n,adj=adj))
}
xy <- c(50,50)
out <- my.rect.lattice(xy)
plot(out$X,pch=20,cex=0.001)

Connect two squares for example with only one shared vertex

X2 <- rbind(out$X,out$X + 50)
plot(X2)

adj <- matrix(0,out$n*2-1,out$n*2-1)
adj[1:out$n,1:out$n] <- out$adj
adj[out$n:(2*out$n-1),out$n:(2*out$n-1)] <- out$adj
eigen.out <-eigen(adj)

p <- 1 
col <- my.col.st(eigen.out[[2]][,1],N=5)
plot(X2,pch=20,cex=0.0001)
my.color.point(X2,col,cex=0.1)

Plot series of eigen vectors

for(i in 1:20){
  col <- my.col.st(eigen.out[[2]][,i],N=5)
  plot(X2,pch=20,cex=0.0001,main=paste(i, "-st"))
  my.color.point(X2,col,cex=0.1)
}